Raised by Einstein et al. [1]} in discussions on the
incompleteness of quantum mechanics in 1935, quantum nonlocality
[2]} has been extensively studied and played significant
roles in a variety of quantum tasks [3]}. Recent studies have
shown that quantum nonlocality is also a unique quantum resource for
some device-independent quantum tasks such as the key distribution
[4]}, random expansion [5]}, random amplification
[6]} and the related experiments [7]}, [8]}, [9]}. In
particular, in 1987 a special kind of nonlocality called genuine tripartite nonlocality (GTN) was introduced by Svetlichny for tripartite systems [10]}.
Such multipartite nonlocal correlations have not only foundational
implications [11]} but also novel applications in quantum
communication and quantum computation [12]}, [13]}, [14]}, [15]}, as
well as in phase transitions and criticality in many-body systems
[14]}. GTN can be detected by the violation of
Svetlichny inequality. Let \(A_i=\vec{a}_i\cdot \vec{\sigma }\) ,
\(B_i=\vec{b}_i\cdot \vec{\sigma }\)  and
\(C_i=\vec{c}_i\cdot \vec{\sigma }\) , \(i=0,1\) , be the measurement
observables associated with the first, second and third qubit, respectively, with
\(\vec{a}_i\) , \(\vec{b}_i\)  and \(\vec{c}_i\)  the real unit vectors,
\(\vec{\sigma }=(\sigma _1,\sigma _2,\sigma _3)\)  the vector given by the standard
Pauli matrices. The Svetlichny operator is defined by
\(& &\mathbb {S}=A_0(B_0+B_1)C_0+A_0(B_0-B_1)C_1\nonumber \\& &+A_1(B_0-B_1)C_0-A_1(B_0+B_1)C_1.\) 
